# What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?

Let $$L$$ be a number field, $$\mathcal{O}_L$$ its ring of integers, and $$\mathcal{Cl(O}_L)$$ its ideal class group. Let's fix an arbitrary class $$[c] \in \mathcal{Cl(O}_L)$$. By $$r(n)=r([c], n)$$, I mean the number of ideals of norm $$n$$, that belong to the class $$[c]$$,

$$r(n)=r([c], n)= \sharp\bigg\{ \mathfrak{I} \subseteq \mathcal{O}_L: \mathfrak{I} \in [c], N(\mathfrak{I})=n \bigg\}.$$

Dedekind zeta function, is something which is related to all of these $$r([c], n)$$'s, where $$[c]$$ varies arbitrarily in $$\mathcal{Cl(O}_L)$$. I am curious about the situation when we restrict ourselves to only one arbitrary but fixed class $$[c] \in \mathcal{Cl(O}_L)$$: What are the "well-known" functions in number theory, that are related to $$r(n)=r([c], n)$$'s?

What I know: Suppose that $$L$$ is an imaginary quadratic field, $$D$$ its discriminant, and let $$[c]$$ be an arbitrary but fixed class in $$\mathcal{Cl(O}_L)$$, and let $$\Theta(z)=\Theta_{[c]}(z)= 1+\sum_{n=1}^{\infty} r(n)q^n$$, where $$q=e^{2\pi i z}$$, and $$r(n)=r([c], n)$$. Then $$\Theta(z)$$ is a modular form of weight $$1$$, with level $$N=\vert D \vert$$ and charachter $$\chi(.)=\left(\dfrac{D}{.}\right)$$; i.e. for $$z \in \mathcal{H}$$ and $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$$ we have $$\Theta\left(\dfrac{az+b}{cz+d}\right) = \chi(a)\left(cz+d\right) \Theta(z)$$. [We can associate a quadratic form the class $$[c]$$, and here $$\Theta(z)$$ is the associated theta series.]

It sounds like you're looking for something like the function $$\zeta_C(s) = \sum_{\mathfrak{a} \in C} N(\mathfrak{a})^{-s} = \sum_{n \ge 1} r([c], n) n^{-s}.$$ These functions are sometimes called "ideal class zeta functions" and they come up from time to time in the literature. See e.g. this paper in J London Math Soc:
The related functions where you sum over a character $$\psi$$ of the ideal class group, $$L(\psi, s) = \sum_{C \in Cl(L)} \psi(C) \zeta_C(s),$$ are much more commonly studied (since they have an Euler product expansion, which isn't true of the $$\zeta_C(s)$$ individually). These are examples of Hecke $$L$$-functions.
• $L(\psi, s) = \prod_{\mathfrak{P}}(1 - \psi([\mathfrak{P}]) N(\mathfrak{P})^{-s})^{-1}$ (product over prime ideals of $O_L$). Jan 9, 2021 at 17:56